Abstract
We study the task of entanglement distillation in the one-shot setting under different classes of quantum operations which extend the set of local operations and classical communication (LOCC). Establishing a general formalism which allows for a straightforward comparison of their exact achievable performance, we relate the fidelity of distillation under these classes of operations with a family of entanglement monotones and the rates of distillation with a class of smoothed entropic quantities based on the hypothesis testing relative entropy. We then characterise exactly the one-shot distillable entanglement of several classes of quantum states and reveal many simplifications in their manipulation. We show in particular that the ε-error one-shot distillable entanglement of any pure state is the same under all sets of operations ranging from one-way LOCC to separability-preserving operations or operations preserving the set of states with positive partial transpose, and can be computed exactly as a quadratically constrained linear program. We establish similar operational equivalences in the distillation of isotropic and maximally correlated states, reducing the computation of the relevant quantities to linear or semidefinite programs. We also show that all considered sets of operations achieve the same performance in environment-assisted entanglement distillation from any state.
Highlights
Quantum entanglement plays a fundamental role in quantum information processing by serving as a resource which underlies many important protocols such as quantum teleportation [1] or superdense coding [2] as well as quantum technological applications such as quantum repeaters and networks [3, 4]
We develop a comprehensive framework for the study of one-shot entanglement distillation under several different classes of operations — separable maps (SEP), separability-preserving maps (SEPP), positive partial transpose (PPT) maps, two types of positive partial transpose–preserving maps, as well as two types of maps based on the so-called Rains set — many of which have been considered in the literature as a relaxation of local operations and classical communication (LOCC) in various contexts, but whose one-shot distillation capabilities in relation to other operations remained unknown
Such extensions are still bound by operationally motivated constraints (e.g., SEPP can never generate entanglement from an unentangled state, just as LOCC), but they can often be understood as allowing for additional resources to be used in entanglement manipulation
Summary
Quantum entanglement plays a fundamental role in quantum information processing by serving as a resource which underlies many important protocols such as quantum teleportation [1] or superdense coding [2] as well as quantum technological applications such as quantum repeaters and networks [3, 4]. We develop a comprehensive framework for the study of one-shot entanglement distillation under several different classes of operations — separable maps (SEP), separability-preserving maps (SEPP), positive partial transpose (PPT) maps, two types of positive partial transpose–preserving maps, as well as two types of maps based on the so-called Rains set — many of which have been considered in the literature as a relaxation of LOCC in various contexts, but whose one-shot distillation capabilities in relation to other operations remained unknown Such extensions are still bound by operationally motivated constraints (e.g., SEPP can never generate entanglement from an unentangled state, just as LOCC), but they can often be understood as allowing for additional resources to be used in entanglement manipulation (e.g, any PPT operation can be stochastically implemented by LOCC if one is given access to a bound entangled state [28]). Our work improves many earlier results in the characterisation of one-shot entanglement distillation [16, 17, 19, 20, 29], which relied on approximate bounds and were only exact asymptotically; crucially, our formalism allows for a precise description of distillation already at the one-shot level, providing an exact characterisation of the operational power of several classes of operations which extend LOCC and shedding light on the capabilities of LOCC themselves
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